AQL/RQL/LTPD/OC Curve/Reliability and Confidence:

It has been a while since I have posted about statistics. In today’s post, I am sharing a spreadsheet that generates an OC Curve based on your sample size and the number of rejects. I get asked a lot about a way to calculate sample sizes based on reliability and confidence levels. I have written several posts before. Check this post and this post for additional details.

The spreadsheet is hopefully straightforward to use. The user has to enter data in the required yellow cells.

A good rule of thumb is to use 95% confidence level, which also corresponds to 0.05 alpha. The spreadsheet will plot two curves. One is the standard OC curve, and the other is an inverse OC curve. The inverse OC curve has the probability of rejection on the Y-axis and % Conforming on the X-axis. These corresponds to Confidence level and Reliability respectively.

I will discuss the OC curve and how we can get a statement that corresponds to a Reliability/Confidence level from the OC curve.

The OC Curve is a plot between % Nonconforming, and Probability of Acceptance. Lower the % Nonconforming, the higher the Probability of Acceptance. The probability can be calculated using Binomial, Hypergeometric or Poisson distributions. The OC Curve shown is for n = 59 with 0 rejects calculated using Binomial Distribution.

The Producer’s risk is the risk of good product getting rejected. The Acceptance Quality Limit (AQL) is generally defined as the percent defectives that the plan will accept 95% of the time (in the long run). Lots that are at or better than the AQL will be accepted 95% of the time (in the long run). If the lot fails, we can say with 95% confidence that the lot quality level is worse than the AQL. Likewise, we can say that a lot at the AQL that is acceptable has a 5% chance of being rejected. In the example, the AQL is 0.09%.

The Consumer’s risk, on the other hand, is the risk of accepting bad product. The Lot Tolerance Percent Defective (LTPD) is generally defined as percent defective that the plan will reject 90% of the time (in the long run). We can say that a lot at or worse than the LTPD will be rejected 90% of the time (in the long run). If the lot passes, we can say with 90% confidence that the lot quality is better than the LTPD (% nonconforming is less than the LTPD value). We could also say that a lot at the LTPD that is defective has a 10% chance of being accepted.

The vertical axis (Y-axis) of the OC Curve goes from 0% to 100% Probability of Acceptance. Alternatively, we can say that the Y-axis corresponds to 100% to 0% Probability of Rejection. Let’s call this Confidence.

The horizontal axis (X-axis) of the OC Curve goes from 0% to 100% for % Nonconforming. Alternatively, we can say that the X-axis corresponds to 100% to 0% for % Conforming. Let’s call this Reliability.

We can easily invert the Y-axis so that it aligns with a 0 to 100% confidence level. In addition, we can also invert the X-axis so that it aligns with a 0 to 100% reliability level. This is shown below.

What we can see is that, for a given sample size and defects, the more reliability we try to claim, the less confidence we can assume. For example, in the extreme case, 100% reliability lines up with 0% confidence.

I welcome the reader to play around with the spreadsheet. I am very much interested in your feedback and questions. The spreadsheet is available here.

In case you missed it, my last post was Nature of Order for Conceptual Models: